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The analysis of piezomagnetoelastic energy harvesters under broadbandrandom excitations
S. F. Ali,1,a) S. Adhikari,1,b) M. I. Friswell,1 and S. Narayanan2
1College of Engineering, Swansea University, Singleton Park, Swansea SA2 8PP, United Kingdom2Indian Institute of Technology Madras, Chennai, India
(Received 13 January 2011; accepted 29 January 2011; published online 6 April 2011)
This paper presents an analysis of piezomagnetoelastic energy harvesters under broadband random
ambient excitations for the purpose of powering low-power electronic sensor systems. Their
nonlinear behavior as a result of vibration in a magnetic field makes piezomagnetoelastic energy
harvesters different from classical piezoelastic energy harvesters. An equivalent linearization-based
analytical approach is developed for the analysis of harvested power. A closed-form approximate
expression for the ensemble average of the harvested power is derived and validated against
numerical Monte Carlo simulation results. Our results show that it is possible to optimally design the
system such that the mean harvested power is maximized for a given strength of the input broadband
random ambient excitation. VC 2011 American Institute of Physics. [doi:10.1063/1.3560523]
I. INTRODUCTION
The evolution of electronic devices since the beginning
of micro and nanoelectronics has brought about exponential
growth in computational power in ever-shrinking systems;
the associated increase in functionality brings a revolution in
systems targeting wearable healthcare, lifestyle, and indus-
trial monitoring applications.
As systems continue to shrink, less energy is required
onboard. This has allowed another energy paradigm, namely
energy harvesting (also referred to as energy scavenging) from
the environment. A decade of research in the field of energy
harvesting has led to efficient capture of small amounts of
energy from the environment and their transformation into
electrical energy. Energy-autonomous systems using energy
harvesting are particularly attractive when long-term remote
deployment is needed or wherever a natural long-term energy
source is available (such as temperature or vibrations) for con-
tinuous replenishment of energy consumed by the system. Such
an inexhaustible energy supply is a significant advantage over a
battery supply or mains power. Extended lifetime and autonomy
are particularly advantageous in systems with limited accessibil-
ity, such as medical implants and infrastructure-integrated
micro-sensors, wireless sensor nodes used for structural health
monitoring, embedded and implanted sensor nodes for medical
applications, recharging the batteries of large systems, monitor-
ing tire pressure in automobiles, powering unmanned vehicles,
and running household security systems.
As described by Williams and Yates,1 the three basic
vibration-to-electric energy conversion mechanisms are elec-
tromagnetic,1–4 electrostatic,5,6 and piezoelectric7–10 transduc-
tions. In the last decade, these transduction mechanisms have
been investigated by numerous researchers for vibration-based
energy harvesting and extensive discussions can be found in
existing review articles.5,9,11,12
Regardless of the transduction mechanism, a primary issue
in vibration-based energy harvesting is that the best perform-
ance of a generator is usually limited to excitation at its funda-
mental resonance frequency. If the applied ambient vibration
deviates slightly from the resonance condition then power out-
put is drastically reduced. Hence, a major issue in energy har-
vesting is to enable broadband energy harvesters.13,14 Thus,
researchers have recently focused on the concept of broadband
energy harvesting to solve this issue with different approaches.
The most recent addition to broadband energy harvesters
is piezomagnetoelastic systems.14 This magnetoelastic struc-
ture (described in the following section) was first investi-
gated by Moon and Holmes15 as a mechanical structure that
exhibits strange attractor motions. Erturk et al.14 investigated
the potential of this device for energy harvesting when the
excitation is harmonic and demonstrated an order of magni-
tude larger power output over the linear system (without
magnets) for nonresonant excitation. Litak et al.16 consid-
ered the performance of nonlinear piezomagnetoelastic sys-
tem under random excitations using numerical methods.
The theoretical analysis of piezomagnetoelastic systems
is absent in the literature. Analysis of a nonlinear piezomagne-
toelastic energy harvesting system under random excitations
requires solution of the multidimensional Fokker–Planck
equation to obtain the governing probability density function
of the harvested power. This paper considers a different
approach and stochastically linearizes the system to develop
an equivalent model of the overall system. The linear model is
then analyzed to determine probability density functions of
the system response and the power scavenged by the system.
The paper is organized as follows: The next section describes
the piezomagnetoelastic harvester. Details of the stochastic
linearization technique are also given. A brief description of
stochastic random vibration theory and the three nonlinear
equations that must be solved to obtain the equivalent linear
system are reported in Sec. III. The equivalent linear system
under zero-mean white noise excitation is analyzed in
Sec. IV. Numerical results are reported in Sec. V.
a)Electronic mail: sk.faruque.ali@gmail.com.b)Electronic mail: s.adhikari@swansea.ac.uk.
0021-8979/2011/109(7)/074904/8/$30.00 VC 2011 American Institute of Physics109, 074904-1
JOURNAL OF APPLIED PHYSICS 109, 074904 (2011)
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp
II. PIEZOMAGNETOELASTIC ENERGY HARVESTERS
The device consists of a ferromagnetic cantilever beam
that is excited at the support (see Fig. 1). Two permanent
magnets are located symmetrically on the base near the free
end. The distance between the beam and the magnets deter-
mines the stable equilibrium points. Here we are interested
in the case when the system has three equilibrium positions,
two of which are stable, and the mechanical system is char-
acterized by the classical double-well potential. The nondi-
mensional equations of motion for this system are
€xþ 2f _x� 1
2xð1� x2Þ � vv ¼ f ðtÞ; (1)
_vþ kvþ j _x ¼ 0; (2)
where x is the dimensionless transverse displacement of the
beam tip, t is the dimensionless voltage across the load resis-
tor, v is the dimensionless piezoelectric coupling term in the
mechanical equation, j is the dimensionless piezoelectric cou-
pling term in the electrical equation, k / 1=RlCp is the recip-
rocal of the dimensionless time constant of the electrical
circuit, Rl is the load resistance, and Cp is the capacitance of
the piezoelectric material. The force f ðtÞ is proportional to the
base acceleration on the device. A simulation-based nonlinear
analysis for random base excitation is reported by the authors
in Ref. 16. Gaussian white noise excitation, with zero mean
and specified variance, was assumed for the simulations. The
study demonstrated that the dynamics of the system is a dou-
ble-well potential and the greatest energy generation occurs
when the system moves between the potential wells.
Equation (1) is a nonlinear equation with nonlinearity in
the stiffness term, whereas Eq. (2) is a linear equation. To
facilitate the linearization process, Eq. (1) can be rewritten as
€xþ 2f _xþ gðxÞ � vv ¼ f ðtÞ: (3)
The nonlinear stiffness is represented as gðxÞ ¼ �12ðx�x3Þ.
This paper will develop a linearized model for the Duffing
equation [Eq. (3)] based on the stochastic linearization
approach.17
Assuming a nonzero-mean random excitation [i.e., f ðtÞ¼ f0ðtÞ þ mf ] and a nonzero-mean system response [i.e.,
xðtÞ ¼ x0ðtÞ þ mx], the following equivalent linear system is
considered,
€x0 þ 2f _x0 þ a0x0 þ b0 � vt ¼ f0ðtÞ þ mf (4)
where f0ðtÞ and x0ðtÞ are zero-mean random processes, mf
and mx are the mean of the original processes f ðtÞ and xðtÞrespectively, a0 and b0 are the constants to be determined,
with b0 ¼ mf and a0 the square of the natural frequency of
the linearized system x2eq. Stochastic moments of the nonlin-
ear and the linearized model outputs will be matched by
minimizing the expectation of the error norm, i.e.,
(E e2½ �;with e¼gðxÞ�a0x0�b0). To determine the constants
a0 and b0 in terms of the statistics of the response x, we take
partial derivatives of the error norm with respect to a0 and b0
and equate them to zero individually.
@
@a0
E½e2� ¼ E½gðxÞx0� � a0E½x20� � b0E½x0� (5)
@
@b0
E½e2� ¼ E½gðxÞ� � a0E½x0� � b0: (6)
Equating Eqs. (5) and (6) to zero, we get,
a0 ¼E½gðxÞx0�
E½x20�
¼ E½gðxÞx0�r2
x
(7)
b0 ¼ E½gðxÞ�:¼ mf (8)
As a special case, if we assume that xðtÞ is a Gaussian ran-
dom process, the expressions in Eqs. (7) and (8) can be fur-
ther simplified17,18 as,
a0 ¼ E
�d
dxgðxÞ
�
¼ � 1
21� 3E½x2�� �
¼ � 1
21� 3r2
x � 3m2x
� �(9)
which gives
3m2x þ 3r2
x � 2a0 � 1 ¼ 0 (10)
and
mf ¼ E½gðxÞ�
¼ � 1
2E½x� � E½x3�� �
:(11)
Note that for a nonzero-mean Gaussian process x with mean
mx and its zero-mean part x0, we have the following
relations.
E½x30� ¼ E½ðx� mxÞ3� ¼ E½x3� � 3mxr
2x � m3
x : (12)
FIG. 1. Schematic diagram of magnetopiezoelastic device (Ref. 14). Reprinted
with permission from Appl. Phys. Lett. 94, 254102 (2009). Copyright 2009,
American Institute of Physics.
074904-2 Ali et al. J. Appl. Phys. 109, 074904 (2011)
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp
Since for a zero-mean Gaussian process E½x30� ¼ 0, using the
relation in Eq. (12) we get,
E½x3� ¼ 3mxr2x þ m3
x : (13)
Substituting E½x3� from Eq. (13) into Eq. (11) we get
mf ¼ �1
2mx � 3mxr
2x þ m3
x
� �� �(14)
which gives
mx m2x þ 3r2
x � 1� �
� 2mf ¼ 0 (15)
where mx and rx are the mean and standard deviation of the
system response x.
The process of statistical linearization reduces to finding
three unknowns a0, mx and rx with only two equations [Eqs.
(10) and (15)]. Another expression for rx can be obtained
from the linearized system equations.
Eq. (4) along with Eq. (2) can be rewritten as
€x0 þ 2f _x0 þ a0x0 � vt ¼ f0ðtÞ (16)
_vþ ktþ j _x ¼ 0: (17)
Taking the Fourier transform of Eqs. (16) and (17), we get
a0 � X2� �
þ 2iXf �viXj iXþ kð Þ
� �XðXÞVðXÞ
� ¼ F0ðXÞ
0
� :
(18)
Inverting the coefficient matrix, the displacement and volt-
age in the frequency domain can be obtained as
XV
� ¼ 1
D
iXþ kð Þ v
�iXj a0 � X2� �
þ 2iXf
" #F0
0
� (19)
¼ HðXÞ F0
0
� (20)
where HðXÞ is the 2� 2 matrix of frequency-response func-
tions and the determinant of the coefficient matrix is
DðiXÞ ¼ ðiXÞ3þ 2fþ kð ÞðiXÞ2þ 2fkþ jvþ a0ð ÞðiXÞ þ ka0:
(21)
III. A BRIEF OVERVIEW OF STATIONARY RANDOMVIBRATION
We consider the excitation f0ðtÞ to be a zero-mean,
weakly stationary, Gaussian, broadband random process. To
obtain statistics of random response quantities such as dis-
placement of the mass xðtÞ and the voltage vðtÞ, one needs to
solve the coupled stochastic differential equations [Eqs. (16)
and (17)]. However, analytical results developed within the
theory of random vibration allow us to bypass numerical sol-
utions because we are interested in the average values of the
output random processes.
Since f0ðtÞ is a weakly stationary random process, its
autocorrelation function depends only on the difference in
the time instants, and thus
E½f0ðs1Þf0ðs2Þ� ¼ Rf0f0ðs1 � s2Þ: (22)
This autocorrelation function can be expressed as the inverse
Fourier transform of the spectral density Uf0f0ðXÞ as
Rf0f0ðs1 � s2Þ ¼ð1�1
Uf0f0ðXÞ exp½iXðs1 � s2Þ�dX: (23)
For a damped linear system of the form shown in Eq. (20), it
can be shown that17,19 the spectral density of XðXÞ is related
to the spectral density of the excitation F0 by
Ux0x0ðXÞ ¼ jH11ðXÞj2Uf0f0ðXÞ: (24)
where H11ðXÞ is the element in the first row and column of
the matrix HðXÞ. In this paper we are interested in the stand-
ard deviation of the response, rx, given as
r2x ¼ r2
x0 ¼ð1�1
H11ðXÞ2Uf0f0 dX (25)
where Uf0f0 is a constant for the weakly stationary, white
noise process.
Combining Eqs. (19), (20), (21), and (25), we get
r2x ¼ Uf0f0
ð1�1
k2 þ X2
DðXÞD�ðXÞ dX: (26)
The calculation of the integral on the right-hand side of Eq.
(26) in general requires the calculation of integrals of the fol-
lowing form
In ¼ð1�1
NnðXÞ dXKnðXÞK�nðXÞ
(27)
where the polynomials have the form
NnðxÞ ¼ pn�1X2n�2 þ pn�2X
2n�4 þ � � � þ p0 (28)
KnðxÞ ¼ qnðiXÞn þ qn�1ðiXÞn�1 þ � � � þ q0: (29)
Following Roberts and Spanos17 and Adhikari et al.20 this in-
tegral can be evaluated as
In ¼pqn
det Nn½ �det Dn½ �
: (30)
Here the m� m matrices are defined as
Nn ¼
pn�1 pn�2 � � � p0
�qn qn�2 �qn�4 qn�6 � � � 0 � � �0 �qn�1 qn�3 �qn�5 � � � 0 � � �0 qn �qn�2 qn�4 � � � 0 � � �0 � � � � � � 0 � � �0 0 � � � �q2 q0
26666664
37777775
(31)
074904-3 Ali et al. J. Appl. Phys. 109, 074904 (2011)
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and
Dn ¼
qn�1 �qn�3 qn�5 �qn�7
�qn qn�2 �qn�4 qn�6 � � � 0 � � �0 �qn�1 qn�3 �qn�5 � � � 0 � � �0 qn �qn�2 qn�4 � � � 0 � � �0 � � � � � � 0 � � �0 0 � � � �q2 q0
26666664
37777775:
(32)
Comparing Eqs. (21) and (26) with the general integral in
Eqs. (27) and (28) we have
n ¼ 3;
p2 ¼ 0; p1 ¼ 1; p0 ¼ k2; and
q3 ¼ 1; q2 ¼ 2fþ kð Þ; q1 ¼ 2kfþ jvþ a0ð Þ; q0 ¼ a0k:
(33)
Now using Eq. (33), the integral in Eq. (30) becomes,
In ¼pða0 þ 2kfþ k2Þ
a0 4kf2 þ 2fjvþ 2fa0 þ 2k2fþ kjv� � : (34)
Combining Eq. (34) with Eq. (26) and then simplifying the
resulting expression we obtain the final relation between a0
and rx, which is given as
a0r2x 4kf2 þ 2fjvþ 2fa0 þ 2k2fþ kjv� �� pUf0f0 a0 þ 2kfþ k2
� �¼ 0: (35)
Equation (35) along with Eqs. (10) and (15) provide three
equations to solve for the unknown variables a0, rx, and
mx. Analytical solutions of Eqs. (10), (15), and (35) are not
possible and one should make use of numerical schemes.
In the next section, solutions for zero-mean white noise
excitation is shown. To summarize, one has to solve for mx,
rx, and a0 from the following three nonlinear coupled
equations,
3m2x þ 3r2
x � 2a0 � 1 ¼ 0
mx m2x þ 3r2
x � 1� �
� 2mf ¼ 0
a0r2x 4kf2 þ 2fjvþ 2fa0 þ 2k2fþ kjv� ��pUf0f0 a0 þ 2kfþ k2
� �¼ 0:
(36)
IV. ZERO–MEAN WHITE NOISE EXCITATION
A. Determination of rx , a0, and mx
Without loss of generality the external excitation can be
assumed to be a zero-mean white noise process, i.e., mf ¼ 0.
This largely simplifies our analysis and provides a simple
relation between the mean (mx) and the standard deviation
(rx) of the response. Putting mf ¼ 0 in Eq. (15) we get,
mx m2x þ 3r2
x � 1� �
¼ 0 (37)
which gives either mx ¼ 0 or mx ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 3r2
x
p:
Substituting mx ¼ 0 in Eq. (10) we get
3r2x ¼ 1þ 2a0: (38)
Equation (38) shows that for mx ¼ 0 and for any real a0 � 0,
we have r2x � 1
3.
Substituting mx ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 3r2
x
pin Eq. (10) we get
3ð1� 3r2xÞ þ 3r2
x � 2a0 � 1 ¼ 0;
1� 3r2x � a0 ¼ 0 (39)
which gives
3r2x ¼ 1� a0: (40)
Equation (40) shows that for m2x ¼ 1� 3r2
x and real, positive
a0, we have r2x 2 ½0; 1
3�.
This, on the other hand, bounds a0 2 ½0; 1�. A further
analysis will show that a0 ¼ m2x .
Substituting Eq. (38) and Eq. (40) separately in Eq. (35)
we obtain the following two cubic polynomial equations
respectively.
4fa30 þ 8kf2 þ 4k2fþ 4jvfþ 2fþ 2kjv
� �a2
0
þ 4kf2 þ 2k2fþ 2jvfþ kjv� 3Uf0f0p� �
a0
� 3Uf0f0pk kþ 2fð Þ ¼ 0 (41)
2fa30 þ 4kf2 þ 2k2fþ 2jvf� 2fþ kjv
� �a2
0
� 4kf2 þ 2k2fþ 2jvfþ kjv� 3Uf0f0p� �
a0
þ 3Uf0f0pk kþ 2fð Þ ¼ 0: (42)
Numerical solutions show that Eq. (41) has a single real pos-
itive root and Eq. (42) has two real positive roots for any
given value of Uf0f0 .
B. Determination of E ½t2�
The spectral density of the voltage generated across the
harvester can be related to the excitation as
UvvðXÞ ¼ H21ðXÞ2�� ��Uf0f0ðXÞ; (43)
where H21ðXÞ is the element in the second row and first col-
umn of the matrix HðXÞ [see Equation (19)].
E½t2� ¼ð1�1
H21ðXÞ2�� ��Uf0f0 dX (44)
where Uf0f0 is a constant for a weakly stationary, white noise
process.
Combining Eqs. (19), (20), (21), and (44), we get
E½t2� ¼ Uf0f0
ð1�1
j2X2
DðXÞD�ðXÞ dX: (45)
Proceeding in a similar way to Sec. III we find an expression
for E½v2� as
E½v2� ¼ j2ka0
a0 4kf2 þ 2fjvþ 2fa0 þ 2k2fþ kjv� � pUf0f0 : (46)
074904-4 Ali et al. J. Appl. Phys. 109, 074904 (2011)
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V. NUMERICAL ANALYSIS
As discussed in Sec. IV A, analytical solutions of Eqs.
(41) and (42) are not feasible. Numerical methods are used
to solve for different values of rx, a0, and mx. The solutions
of Eqs. (41) and (42) for various values of standard deviation
of the excitation are reported.
The system parameters have been taken as follows:14
f ¼ 0:01, v ¼ 0:05, and j ¼ 0:5, while k varied between
0:01 and 0:05. The excitation f ðtÞ is considered stationary
Gaussian white noise with standard deviation rf ranging
from 1% to 11%. Higher values of standard deviation are not
considered, as statistical linearization would fail to represent
the system for higher standard deviations.
The nonlinear piezomagnetoelastic system given by
Eq. (1) represents a Duffing-type equation of motion. The
nonlinear system has three equilibrium points, two stable
equilibrium points at x ¼ 61 and one unstable equilibrium
point at the origin (x ¼ 0). Numerical simulations show that
Eq. (41) has a single real positive solution for a0, whereas
Eq. (42) has two real positive solutions. The solutions for
different standard deviations of excitation are shown in Fig.
2. Solution-1 and Solution-2 are the solutions for Eq. (42),
i.e., where mx ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 3r2
x
p, and Solution-3 is obtained by
solving Eq. (41), i.e., the solution representing the zero-
mean response. This representation of the solutions is the
same for all figures reported in this article.
Figures 2(a) and 2(b) are for k ¼ 0:01 and k ¼ 0:05,
respectively. Note that a0 is the square of the natural fre-
quency of the equivalent linear system [see Equation (4)]
and therefore cannot be negative. The blue curve (with
square markers) represents both of the x ¼ 61 stable equi-
librium points. The black dotted line represents the solution
a0 ¼ 0 up to rf ¼ 0:04 and rf ¼ 0:05 for k ¼ 0:01 and
k ¼ 0:05 respectively. This solution is the unstable equilib-
rium point of the system. Therefore the only practical solu-
tion for the system below rf ¼ 0:04 for electrical constant
k ¼ 0:01 and below rf ¼ 0:05 for k ¼ 0:05 is given by the
blue curve with square markers. At this stage the system
remains in one of the potential wells represented by x ¼ 61.
FIG. 2. (Color online) Square of the natural frequency of the equivalent lin-
ear system for different standard deviations of the excitation (rf ) for (a)
k ¼ 0:01 and (b) k ¼ 0:05. The black dashed line represents unstable solu-
tion at a0 ¼ 0.
FIG. 3. (Color online) Square of standard deviation of the response under
different standard deviations of the excitation for (a) k ¼ 0:01 and (b)
k ¼ 0:05. The black dashed line represents unstable solution at a0 ¼ 0.
074904-5 Ali et al. J. Appl. Phys. 109, 074904 (2011)
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When the excitation standard deviation reaches the cut
off (rf ¼ 0:04 for k ¼ 0:01 and rf ¼ 0:05 for k ¼ 0:05) the
system has enough energy to cross the potential barrier and
could jump to the other potential well. This jump from one
potential well to another generates more energy, as will be
shown later. These cut-off standard deviations of the excita-
tion are explained as stochastic resonance phenomena in
Litak et al.16
Figure 3 shows the square of the standard deviation of
the response versus the standard deviation of the excitation.
It can be observed (and has been described in Sec. IV A) that
two real positive solutions of r2x exist for mx 6¼ 0. which
have values less than 13. Another real positive solution is
obtained by solving Eq. (41), which results in rx � 13.
Note that the square of the standard deviation of the
response r2x ¼ 1
3for Solution-2 and Solution-3, as long as the
standard deviation of excitation rf � 0:04 and rf � 0:05 for
k ¼ 0:04 and k ¼ 0:05, respectively. At the cut-off standard
deviation of excitation the standard deviation of response
could jump from Solution-1 to Solution-2 or Solution-3
(depending on the initial condition of the system), generating
more power. This phenomenon is observed in the solution of
the nonlinear system as shown in Litak et al.16 Another point
to note is that the r2x curves are similar for both k ¼ 0:04 and
k ¼ 0:05, the only distinction being the cut-off value of rf .
This shows that the electrical constant (k) plays little role in
the response of the system and that the mechanical and elec-
trical systems are only weakly coupled.
Figure 4 shows the ratio of standard deviations of the
response and the excitation. An order of magnitude increase
in the standard deviation of the response is seen near the
cut-off region. Figure 5 shows the variation of the square of
the mean of the system responses for different values of rf .
As described earlier, the third solution is the zero-mean
response. Interestingly the mean responses seem to be con-
verging to a particular value with increase in standard devia-
tion of excitation. As the excitation noise is increased the
oscillation of the magnetoelastic rod crosses the potential
barrier between individual equilibria and keeps oscillating.
Therefore the equivalent linearization does not represent any
equilibrium value at higher excitation deviations, rather it
shows a new point over which the system keeps oscillating.
The variance of voltage, which is a measure of the mean
power harvested, is shown in Fig. 6 for k ¼ 0:01 and
k ¼ 0:05. Figures 6(a) and 6(b) show that the voltage
increases across the circuit near the cut-off values of the
FIG. 4. (Color online) Ratio of standard deviation of displacement to stand-
ard deviation of the excitation under different standard deviations of the ex-
citation for (a) k ¼ 0:01 and (b) k ¼ 0:05. The black dashed line represents
the unstable solution at a0 ¼ 0.
FIG. 5. (Color online) Square of the mean of the displacement response
against the excitation standard deviation for (a) k ¼ 0:01 and (b) k ¼ 0:05.
074904-6 Ali et al. J. Appl. Phys. 109, 074904 (2011)
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excitation standard deviation. Note that the power remains
very low for noise intensities below the cut-off value. The
electrical constant k significantly affects the voltage pro-
duced and, hence, the power generated. Note that this
increase in power above the cut-off point is a result of
changes in the electrical system, since the mechanical
responses shown in Figs. 2–5 are almost identical. The prac-
tical implementation of a increase in k requires a reduction
in either electrical resistance or capacitance. This matches
the reality, i.e, power increases with a decrease in resistance
across the circuit. These requirements should be included in
trade-off studies required for the design of a real system.
To validate our equivalent linear model simulation,
results from the linear model are compared to that of the
nonlinear system.16 Simulations for both models were per-
formed 5000 times and then the statistical responses were
compared for different values of the linearized natural fre-
quency at a particular excitation standard deviation. Figure 7
shows the probability density functions of the linear and non-
linear models for the excitation standard deviation rf ¼ 0:10
and k ¼ 0:01. The results do not match exactly, but their
closeness confirms that the equivalent linearization captured
the significant dynamics of the nonlinear piezomagnetoelas-
tic harvesters.
FIG. 7. (Color online) Probability density functions of the square of the
voltage, i.e., mean power across the piezoelectric patch, v2, for rf ¼ 0:1 and
k ¼ 0:01. (a) Solution-1, (b) Solution-2, (c) Solution-3.
FIG. 6. (Color online) Variance of voltage against the excitation standard
deviation for (a) k ¼ 0:01 and (b) k ¼ 0:05. The black dashed line represents
unstable solution at a0 ¼ 0.
074904-7 Ali et al. J. Appl. Phys. 109, 074904 (2011)
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp
VI. CONCLUSION
Piezomagnetoelectric harvesters are best suited for
broadband energy harvesting. These devices are nonlinear
and their mechanical counterpart is represented by a Duff-
ing-type oscillator. The device is nonlinear and can have
chaotic motion, depending on the input excitation. Chaotic
motion generates an order of magnitude larger power out-
put over the linear system (without magnets) for nonreso-
nant excitation. This paper develops an equivalent linear
model of the nonlinear system. An analytical expression for
the equivalent linear system is given. The following points
are obtained from the analyses of the equivalent linear
system:
• There exists a cut-off standard deviation of the input exci-
tation, below which the power scavenged by the device is
very low. The power generated by the device increases
rapidly as the standard deviation of the input excitation is
increased.• This cut-off standard deviation of the input excitation
changes with the electrical constant of the system. The
lower the electrical constant (i.e., higher k), the higher the
cut-off standard deviation.• The effect of variation in the electrical constant on the me-
chanical response of the system is relatively small. The
mean and standard deviation of the system displacements
are affected by the change in k only near the cut-off stand-
ard deviation.• The power generating capacity of the system increases
with increasing k. Thus, decreasing the resistance or capac-
itance of the system will increase the harvested power.
This article verifies the equivalent linear system with
results obtained from simulation studies of the nonlinear
system. To have a complete picture of the system behavior
under random excitation one has to solve the Fokker–Planck
equation. This remains for future study.
ACKNOWLEDGMENTS
Dr. S. F. Ali gratefully acknowledges the support of the
Royal Society through a Newton International Fellowship.
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Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp
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